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Fluid Mechanics and Pressure: Understanding Principles and Applications

Explore density, specific gravity, fluid states, pressure, Pascal’s principle, Archimedes’ principle, and more in this lecture on fluid mechanics. Learn how pressure, buoyant force, and flow rate relate to Bernoulli’s equation.

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Fluid Mechanics and Pressure: Understanding Principles and Applications

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  1. PHYS 1443 – Section 002Lecture #21 Wednesday, Dec. 3, 2008 Dr. Jae Yu • Density and Specific Gravity • Fluid and Pressure • Pascal’s Principle • Absolute and Relative Pressure • Buoyant Force and Archimedes’ Principle • Flow Rate and Continuity Equation • Bernoulli’s Equation PHYS 1443-002, Fall 2008 Dr. Jaehoon Yu PHYS 1443-002, Fall 2008 Dr. Jaehoon Yu 1

  2. Announcements • Final exam • Date and Time: 11am – 12:30pm, next Monday, Dec. 8 • Location: SH103 • Comprehensive exam: Covers CH1.1 – CH13.9 + appendices • Mixture of multiple choice and free response problems PHYS 1443-002, Fall 2008 Dr. Jaehoon Yu

  3. Density and Specific Gravity Density, r (rho), of an object is defined as mass per unit volume Unit? Dimension? Specific Gravity of a substance is defined as the ratio of the density of the substance to that of water at 4.0 oC (rH2O=1.00g/cm3). Unit? None Dimension? None Sink in the water What do you think would happen of a substance in the water dependent on SG? Float on the surface PHYS 1443-002, Fall 2008 Dr. Jaehoon Yu

  4. Fluid and Pressure What are the three states of matter? Solid, Liquid and Gas Using the time it takes for a particular substance to change its shape in reaction to external forces. How do you distinguish them? A collection of molecules that are randomly arranged and loosely bound by forces between them or by an external container. What is a fluid? We will first learn about mechanics of fluid at rest, fluid statics. In what ways do you think fluid exerts stress on the object submerged in it? Fluid cannot exert shearing or tensile stress. Thus, the only force the fluid exerts on an object immersed in it is the force perpendicular to the surface of the object. This force by the fluid on an object usually is expressed in the form of the force per unit area at the given depth, the pressure, defined as Expression of pressure for an infinitesimal area dA by the force dF is Note that pressure is a scalar quantity because it’s the magnitude of the force on a surface area A. Special SI unit for pressure is Pascal What is the unit and the dimension of pressure? Unit:N/m2 Dim.: [M][L-1][T-2] PHYS 1443-002, Fall 2008 Dr. Jaehoon Yu

  5. Example for Pressure The mattress of a water bed is 2.00m long by 2.00m wide and 30.0cm deep. a) Find the weight of the water in the mattress. The volume density of water at the normal condition (0oC and 1 atm) is 1000kg/m3. So the total mass of the water in the mattress is Therefore the weight of the water in the mattress is b) Find the pressure exerted by the water on the floor when the bed rests in its normal position, assuming the entire lower surface of the mattress makes contact with the floor. Since the surface area of the mattress is 4.00 m2, the pressure exerted on the floor is PHYS 1443-002, Fall 2008 Dr. Jaehoon Yu

  6. P0A h Mg PA Variation of Pressure and Depth Water pressure increases as a function of depth, and the air pressure decreases as a function of altitude. Why? It seems that the pressure has a lot to do with the total mass of the fluid above the object that puts weight on the object. Let’s imagine the liquid contained in a cylinder with height h and the cross sectional area A immersed in a fluid of density r at rest, as shown in the figure, and the system is in its equilibrium. If the liquid in the cylinder is the same substance as the fluid, the mass of the liquid in the cylinder is Since the system is in its equilibrium Therefore, we obtain The pressure at the depth h below the surface of the fluid open to the atmosphere is greater than the atmospheric pressure by rgh. Atmospheric pressure P0 is PHYS 1443-002, Fall 2008 Dr. Jaehoon Yu

  7. A2 F1 d1 d2 A1 F2 Pascal’s Principle and Hydraulics A change in the pressure applied to a fluid is transmitted undiminished to every point of the fluid and to the walls of the container. What happens if P0is changed? The resultant pressure P at any given depth h increases as much as the change in P0. This is the principle behind hydraulic pressure. How? Since the pressure change caused by the the force F1 applied onto the area A1 is transmitted to the F2 on an area A2. In other words, the force gets multiplied by the ratio of the areas A2/A1 and is transmitted to the force F2 on the surface. Therefore, the resultant force F2 is No, the actual displaced volume of the fluid is the same. And the work done by the forces are still the same. This seems to violate some kind of conservation law, doesn’t it? PHYS 1443-002, Fall 2008 Dr. Jaehoon Yu

  8. Example for Pascal’s Principle In a car lift used in a service station, compressed air exerts a force on a small piston that has a circular cross section and a radius of 5.00cm. This pressure is transmitted by a liquid to a piston that has a radius of 15.0cm. What force must the compressed air exert to lift a car weighing 13,300N? What air pressure produces this force? Using the Pascal’s principle, one can deduce the relationship between the forces, the force exerted by the compressed air is Therefore the necessary pressure of the compressed air is PHYS 1443-002, Fall 2008 Dr. Jaehoon Yu

  9. Example for Pascal’s Principle Estimate the force exerted on your eardrum due to the water above when you are swimming at the bottom of the pool with a depth 5.0 m. We first need to find out the pressure difference that is being exerted on the eardrum. Then estimate the area of the eardrum to find out the force exerted on the eardrum. Since the outward pressure in the middle of the eardrum is the same as normal air pressure Estimating the surface area of the eardrum at 1.0cm2=1.0x10-4 m2, we obtain PHYS 1443-002, Fall 2008 Dr. Jaehoon Yu

  10. H h dy y Example for Pascal’s Principle Water is filled to a height H behind a dam of width w. Determine the resultant force exerted by the water on the dam. Since the water pressure varies as a function of depth, we will have to do some calculus to figure out the total force. The pressure at the depth h is The infinitesimal force dF exerting on a small strip of dam dy is Therefore the total force exerted by the water on the dam is PHYS 1443-002, Fall 2008 Dr. Jaehoon Yu

  11. P0 P h Absolute and Relative Pressure How can one measure pressure? One can measure the pressure using an open-tube manometer, where one end is connected to the system with unknown pressure P and the other open to air with pressure P0. The measured pressure of the system is This is called the absolute pressure, because it is the actual value of the system’s pressure. In many cases we measure the pressure difference with respect to the atmospheric pressure to avoid the effect of the changes in P0 that depends on the environment. This is called gauge or relative pressure. The common barometer which consists of a mercury column with one end closed at vacuum and the other open to the atmosphere was invented by Evangelista Torricelli. Since the closed end is at vacuum, it does not exert any force. 1 atm of air pressure pushes mercury up 76cm. So 1 atm is PHYS 1443-002, Fall 2008 Dr. Jaehoon Yu If one measures the tire pressure with a gauge at 220kPa the actual pressure is 101kPa+220kPa=303kPa.

  12. pinA mg pAA Finger Holds Water in Straw You insert a straw of length L into a tall glass of your favorite beverage. You place your finger over the top of the straw so that no air can get in or out, and then lift the straw from the liquid. You find that the straw strains the liquid such that the distance from the bottom of your finger to the top of the liquid is h. Does the air in the space between your finger and the top of the liquid in the straw have a pressure P that is (a) greater than, (b) equal to, or (c) less than, the atmospheric pressure PA outside the straw? Less What are the forces in this problem? Gravitational force on the mass of the liquid Force exerted on the top surface of the liquid by inside air pressure Force exerted on the bottom surface of the liquid by the outside air Since it is at equilibrium Cancel A and solve for pin So pin is less than PA by rg(L-h). PHYS 1443-002, Fall 2008 Dr. Jaehoon Yu

  13. h Mg B Buoyant Forces and Archimedes’ Principle Why is it so hard to put an inflated beach ball under water while a small piece of steel sinks in the water easily? The water exerts force on an object immersed in the water. This force is called the buoyant force. How large is the buoyant force? The magnitude of the buoyant force always equals the weight of the fluid in the volume displaced by the submerged object. This is called the Archimedes’ principle. What does this mean? Let‘s consider a cube whose height is h and is filled with fluid and in its equilibrium so that its weight Mg is balanced by the buoyant force B. The pressure at the bottom of the cube is larger than the top by rgh. Therefore, Where Mg is the weight of the fluid in the cube. PHYS 1443-002, Fall 2008 Dr. Jaehoon Yu

  14. h Mg B More Archimedes’ Principle Let’s consider the buoyant force in two special cases. Case 1: Totally submerged object Let’s consider an object of mass M, with density r0, is fully immersed in the fluid with density rf . The magnitude of the buoyant force is The weight of the object is Therefore total force in the system is • The total force applies to different directions depending on the difference of the density between the object and the fluid. • If the density of the object is smaller than the density of the fluid, the buoyant force will push the object up to the surface. • If the density of the object is larger than the fluid’s, the object will sink to the bottom of the fluid. What does this tell you? PHYS 1443-002, Fall 2008 Dr. Jaehoon Yu

  15. h Mg B More Archimedes’ Principle Case 2: Floating object Let’s consider an object of mass M, with density r0, is in static equilibrium floating on the surface of the fluid with density rf, and the volume submerged in the fluid is Vf. The magnitude of the buoyant force is The weight of the object is Therefore total force of the system is Since the system is in static equilibrium Since the object is floating, its density is smaller than that of the fluid. The ratio of the densities between the fluid and the object determines the submerged volume under the surface. What does this tell you? PHYS 1443-002, Fall 2008 Dr. Jaehoon Yu

  16. Ex.13-10 for Archimedes’ Principle Archimedes was asked to determine the purity of the gold used in the crown. The legend says that he solved this problem by weighing the crown in air and in water. Suppose the scale read 7.84N in air and 6.86N in water. What should he have to tell the king about the purity of the gold in the crown? In the air the tension exerted by the scale on the object is the weight of the crown In the water the tension exerted by the scale on the object is Therefore the buoyant force B is Since the buoyant force B is The volume of the displaced water by the crown is Therefore the density of the crown is Since the density of pure gold is 19.3x103kg/m3, this crown is not made of pure gold. PHYS 1443-002, Fall 2008 Dr. Jaehoon Yu

  17. Example for Buoyant Force What fraction of an iceberg is submerged in the sea water? Let’s assume that the total volume of the iceberg is Vi. Then the weight of the iceberg Fgi is Let’s then assume that the volume of the iceberg submerged in the sea water is Vw. The buoyant force B caused by the displaced water becomes Since the whole system is at its static equilibrium, we obtain Therefore the fraction of the volume of the iceberg submerged under the surface of the sea water is About 90% of the entire iceberg is submerged in the water!!! PHYS 1443-002, Fall 2008 Dr. Jaehoon Yu

  18. Flow Rate and the Equation of Continuity Study of fluid in motion: Fluid Dynamics If the fluid is water: Hydro-dynamics Water dynamics?? • Streamline or Laminar flow: Each particle of the fluid follows a smooth path, a streamline • Turbulent flow: Erratic, small, whirlpool-like circles called eddy current or eddies which absorbs a lot of energy Two main types of flow Flow rate: the mass of fluid that passes the given point per unit time since the total flow must be conserved Equation of Continuity PHYS 1443-002, Fall 2008 Dr. Jaehoon Yu

  19. Example for Equation of Continuity How large must a heating duct be if air moving at 3.0m/s through it can replenish the air in a room of 300m3 volume every 15 minutes? Assume the air’s density remains constant. Using equation of continuity Since the air density is constant Now let’s imagine the room as the large section of the duct PHYS 1443-002, Fall 2008 Dr. Jaehoon Yu

  20. Bernoulli’s Principle Bernoulli’s Principle: Where the velocity of fluid is high, the pressure is low, and where the velocity is low, the pressure is high. Amount of the work done by the force, F1, that exerts pressure, P1, at point 1 Amount of the work done by the force in the other section of the fluid is Work done by the gravitational force to move the fluid mass, m, from y1 to y2 is PHYS 1443-002, Fall 2008 Dr. Jaehoon Yu

  21. Bernoulli’s Equation cont’d The total amount of the work done on the fluid is From the work-energy principle Since the mass m is contained in the volume that flowed in the motion and Thus, PHYS 1443-002, Fall 2008 Dr. Jaehoon Yu

  22. Since Bernoulli’s Equation cont’d We obtain Re-organize Bernoulli’s Equation Thus, for any two points in the flow Result of Energy conservation! Pascal’s Law For static fluid For the same heights The pressure at the faster section of the fluid is smaller than slower section. PHYS 1443-002, Fall 2008 Dr. Jaehoon Yu

  23. Example for Bernoulli’s Equation Water circulates throughout a house in a hot-water heating system. If the water is pumped at the speed of 0.5m/s through a 4.0cm diameter pipe in the basement under a pressure of 3.0atm, what will be the flow speed and pressure in a 2.6cm diameter pipe on the second 5.0m above? Assume the pipes do not divide into branches. Using the equation of continuity, flow speed on the second floor is Using Bernoulli’s equation, the pressure in the pipe on the second floor is PHYS 1443-002, Fall 2008 Dr. Jaehoon Yu

  24. Congratulations!!!! You all have done very well!!! I certainly had a lot of fun with ya’ll and am truly proud of you! Good luck with your exam!!! Have safe holidays!! PHYS 1443-002, Fall 2008 Dr. Jaehoon Yu

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